Maximal distance for robotic simulation: The convex case

Given two hyper-rectangles inE ⁿ with sides having surface normals in the directions of the axes, each containing a set that touches all 2n sides of its containing hyper-rectangle, it is important to have an easily calculated upper bound on the distance between the sets, for use in a branch and bound algorithm applicable in collision avoidance in robotic simulation. In a previous paper, such a bound was given under the hypothesis that the sets are connected. Here, we consider the case where the sets are convex.The work of the second author was supported in part by the Natural Sciences and Engineering Research Council of Canada. The drawings were prepared by K. Stewart.     

A spectrum result on maximal partial ovoids of the generalized quadrangle , even

This article presents a spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q), q even. We prove that for every integer k in an interval of, roughly, size [q²/10,9q²/10], there exists a maximal partial ovoid of size k on Q(4,q), q even. Since the generalized quadrangle W(q), q even, defined by a symplectic polarity of PG(3,q) is isomorphic to the generalized quadrangle Q(4,q), q even, the same result is obtained for maximal partial ovoids of W(q), q even. As equivalent results, the same spectrum result is obtained for minimal blocking sets with respect to planes of PG(3,q), q even, and for maximal partial 1-systems of lines on the Klein quadric Q⁺(5,q), q even.     

Planar maximal covering location problem under block norm distance measure

This paper introduces a new model for the planar maximal covering location problem (PMCLP) under different block norms. The problem involves locating g facilities anywhere on the plane in order to cover the maximum number of n given demand points. The generalization, in this paper, is that the distance measures assigned to facilities are block norms of different types and different proximity measures. First, the PMCLP under different block norms is modelled as a maximum clique partition problem on an equivalent multi-interval graph. Then, the equivalent graph problem is modelled as an unconstrained binary quadratic problem (UQP). Both the maximum clique partition problem and the UQP are NP-hard problems; therefore, we solve the UQP format through a genetic algorithm heuristic. Computational examples are given.     

A sharp uniqueness result for a class of variational problems solved by a distance function

We consider the minimization problem for an integral functional J, possibly nonconvex and noncoercive in , where is a bounded smooth set. We prove sufficient conditions in order to guarantee that a suitable Minkowski distance is a minimizer of J. The main result is a necessary and sufficient condition in order to have the uniqueness of the minimizer. We show some application to the uniqueness of the solution of a system of PDEs of Monge–Kantorovich type arising in problems of mass transfer theory.     

Weak type estimates for maximal operators with a cylindric distance function

We establish sharp weak-type estimates for the maximal operators T^λ_* associated with cylindric Riesz means for functions on H^p(ℝ³) when 4/5 <p<1 and λ=3/p−5/2, and when p=4/5 and λ>3/p−5/2. The first author was supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD) No. R04-2002-000-20028-0. The third author was supported by a Korea University Grant.     

On the maximum number of permutations with given maximal or minimal distance

Let us denote by R(k, ? λ)[R(k, ? λ)] the maximal number $\text{M}$ such that there exist $\text{M}$ different permutations of the set {1,…, k} such that any two of them have at least λ (at most λ, respectively) common positions. We prove the inequalities R(k, ? λ) ? kR(k ? 1, ? λ ? 1), R(k, ? λ) ? R(k, ? λ ? 1) ? k!, R(k, ? λ) ? kR(k ? 1, ? λ ? 1). We show: R(k, ? k ? 2) = 2, R(k, ? 1) = (k ? 1)!, R(p^m, ? 2) = (p^m ? 2)!, R(p^m + 1, ? 3) = (p^m ? 2)!, $\text{R(k, ? k ? 3) =}\text{k!}\text{2}$, R(k, ? 0) = k, R(p^m, ? 1) = p^m(p^m ? 1), R(p^m + 1, ? 2) = (p^m + 1)p^m(p^m ? 1). The exact value of R(k, ? λ) is determined whenever k ? k₀(k ? λ); we conjecture that R(k, ? λ) = (k ? λ)! for k ? k₀(λ). Bounds for the general case are given and are used to determine that the minimum of |R(k, ? λ) ? R(k, ? λ)| is attained for $\text{λ = (}\text{k}\text{2}\text{) + O(}\text{k}\text{log}\text{k}\text{)}$.     

Definability of Frobenius orbits and a result on rational distance sets

We prove that the first order theory of (possibly transcendental) meromorphic functions of positive characteristic \(p>2\) is undecidable. We also establish a negative solution to an analogue of Hilbert’s tenth problem for such fields of meromorphic functions, for Diophantine equations including vanishing conditions. These undecidability results are proved by showing that the binary relation \(\exists s\ge 0, f=g^{p^s}\) is positive existentially definable in such fields. We also prove that the abc conjecture implies a solution to the Erdös–Ulam problem on rational distance sets. These two seemingly distant topics are addressed by a study of power values of bivariate polynomials of the form F(X)G(Y).     

Mathematical modeling and computer simulation of a robotic rat pup

In this paper, we describe the mathematics and computer implementation of a robotic rat pup simulation. Our goal is to understand neurobehavioral principles in a mammalian model organism—the Norway rat pup (Rattus norvegicus). Our approach is unique in that animal, simulation, and robot studies occur in parallel and inform each other. Behavior is dependent on the nervous system, body morphology, physiology, environment, and the interactions among these elements. Autonomous robotics hardware models and their associated simulations allow the possibility of systematically manipulating variables in each of these elements in ways that would be impossible using live animals. Specifically, we describe the development and validation of a Newtonian-dynamics-based simulation of a robotic rat pup, including mathematical formulation and computer implementation. The computer simulation consists of three distinct components that interact to simulate robotic behavior: (1) dynamics of the robotic rat pup itself, including sensors and actuators, (2) environmental coupling dynamics of the robot arena with the robotic rat pup, and (3) the robot control algorithms as implemented on the physical robot. The mathematical formulation, software implementation, model identification, model validation, and an application example are all described.     

Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model

We are interested in the strong convergence of Euler-Maruyama type approximations to the solution of a class of stochastic differential equations models with highly nonlinear coefficients, arising in mathematical finance. Results in this area can be used to justify Monte Carlo simulations for calibration and valuation. The equations that we study include the Ait-Sahalia type model of the spot interest rate, which has a polynomial drift term that blows up at the origin and a diffusion term with superlinear growth. After establishing existence and uniqueness for the solution, we show that an appropriate implicit numerical method preserves positivity and boundedness of moments, and converges strongly to the true solution.     

A maximal variance problem

In this work, we provide a direct proof concerning a special type of concave density function on a bounded closed interval with minimal variance. This proof involves elementary methods, without using any advanced theories such as Weierstrass’s Approximation Theorem, from which the technical core result of the paper [C. Carlsson, R. Fullér, P. Majlender, On possibilistic correlation, Fuzzy Sets and Systems 155 (2005) 425–445] comes.     

A result of shepp

A simple direct proof is given of a result due to L. Shepp that a certain function of two independent normal random variables with zero means and possibly unequal variances is itself normal.     

A harmonic-type maximal principle in commutant lifting

In this note, we prove a harmonic-type maximal principle for the Schur parametrization of all intertwining liftings of an intertwining contraction in the commutant lifting theorem.     

A family of maximal surfaces in Lorentz-Minkowski three-space

We prove the existence of an infinite family of complete spacelike maximal surfaces with singularities in Lorentz-Minkowski three-space and study their properties. These surfaces are distinguished by their number of handles and have two elliptic catenoidal ends.